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ENERGY
If TO and Do denote the initial values of T and D", then last relation gives
and                   T-T0=-(U-UQ}.                    (
Therefore if only conservative forces act between the vari< parts of an isolated system, the sum of the potential a kinetic energies- of the system remains constant, in ot] words, the gain in the kinetic energy equals the loss in the pot tial energy. Equation (X) will be called the energy equati 167. Conservation of Dynamical Energy and the Law of Act and Reaction. — The principle of the conservation of dyna ical energy may be obtained from the Law of Action a Reaction. In order to prove this statement consider isolated conservative system. Suppose the configuration the system to have changed under the action of its int nal forces. Let Do and U be the potential energies in 1 initial and final configurations, respectively. Then 1 change in the potential energy is
07-Z7o).
During the change in the configuration of the system 1 positions and the velocities of the particles, which form 1 system, undergo changes. Therefore let s0 and s den< the positions, and v0 and v the velocities of any particle the initial and final configurations of the system. Furtl let F denote the resultant force which acts upon the partic Then the change in the potential energy of the system c to the displacement of the particle from s0 to s is
- f/Vds,"
«An
where Fr is the tangential component of the force.   1 normal component contributes nothing to the work.   The
* Potential energy is, by definition, the work done by external forces aga internal forces. Therefore when the change in potential energy is obtained computing the work done by internal forces the result is the negative of change in the potential energy. Hence the negative sign.